Question about a QM postulate

If a system is in state [itex]|\psi>[/itex] before a meassurement, then after we find a value a_n the system is in state
[tex]\frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}}[/tex]

so for instance if [itex]|\psi>=c_1|a_1>+c_2|a_2>+c_3|a_3>[/itex] after we find a_1 the system is in the state [itex]|\psi>'=e^{i\theta}|a_1>[/itex] where [itex]\theta=[/itex] is the argument of complex a_1.
I know that [itex]|\psi>'[/itex] and [itex]|a_1>[/itex] represent two different vectors of the Hilbert space of states but the same physical state.
My question is, is there any way that the phase factor [itex]e^{\theta}[/itex] manifest itself in a physical experiment? or put in another way, does this factor have any physical significance?
I ask this because some systems when rotated 360 degrees transforms like [itex]|a> \rightarrow -|a>[/itex] and this sign can be detected exparimentally

If a system is in state [itex]|\psi>[/itex] before a meassurement, then after we find a value a_n the system is in state
[tex]\frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}}[/tex]

so for instance if [itex]|\psi>=c_1|a_1>+c_2|a_2>+c_3|a_3>[/itex] after we find a_1 the system is in the state [itex]|\psi>'=e^{i\theta}|a_1>[/itex] where [itex]\theta=[/itex] is the argument of complex a_1.

you mean c_1.

I know that [itex]|\psi>'[/itex] and [itex]|a_1>[/itex] represent two different vectors of the Hilbert space of states but the same physical state.
My question is, is there any way that the phase factor [itex]e^{\theta}[/itex] manifest itself in a physical experiment? or put in another way, does this factor have any physical significance?

No, I don't think so.

I ask this because some systems when rotated 360 degrees transforms like [itex]|a> \rightarrow -|a>[/itex] and this sign can be detected exparimentally